Mathematical Option Pricing

Mark H.A.Davs Mathematcal Opton Prcng MSc Course n Mathematcs and Fnance Imperal College London 11 January 26 Department of Mathematcs Imperal College London South Kensngton Campus London SW7 2AZ

Contents 1 Further Results n Stochastc Analyss....................................... 1 1.1 The Martngale Representaton Theorem for Brownan Moton.................. 1 1.2 Changes of Measure....................................................... 3 1.2.1 Normal dstrbutons................................................. 3 1.2.2 A General Settng................................................... 4 1.2.3 Condtonal Expectatons............................................. 4 1.3 The Lévy characterzaton of Brownan Moton................................ 6 1.3.1 Quadratc varaton of Brownan moton................................ 6 1.3.2 Quadratc varaton of contnuous martngales........................... 7 1.3.3 The Lévy characterzaton............................................ 7 1.4 The Grsanov Theorem..................................................... 9 2 The Black Scholes World.................................................... 11 2.1 The Model............................................................... 11 2.2 Portfolos and Tradng Strateges............................................ 11 2.3 Arbtrage and Valuaton.................................................... 12 2.3.1 Forwards........................................................... 12 2.3.2 Put-Call Party..................................................... 12 2.3.3 Replcaton......................................................... 13 2.4 Black-Scholes: the Orgnal Proof............................................ 13 2.4.1 Probablstc soluton of the Black-Scholes PDE......................... 14 2.5 Proof by Martngale Representaton......................................... 15 2.6 Robustness of Black-Scholes Hedgng......................................... 16 2.7 Optons on Dvdend-payng Assets.......................................... 18 2.8 Barrer Optons........................................................... 19 3 Mult-Asset Optons......................................................... 25 3.1 The Margrabe Formula..................................................... 25 3.1.1 The Probablstc Method............................................ 26 3.1.2 Hedgng an Exchange Opton......................................... 27 3.1.3 Exercse Probablty................................................. 28 3.1.4 Margrabe wth dvdends............................................. 28 3.1.5 Black-Scholes as a specal case........................................ 29 3.2 Cross-Currency Optons.................................................... 29 3.2.1 Forward FX rates................................................... 29 3.2.2 The domestc rsk-neutral measure..................................... 29

VI Contents 3.2.3 Opton Valuaton.................................................... 31 3.2.4 Hedgng Quanto Optons............................................. 31 3.3 Numérare pars and change of numérare..................................... 33 3.3.1 Numérare pars..................................................... 34 3.3.2 Change of numérare................................................. 35 3.3.3 Margrabe revsted.................................................. 36 4 Fxed Income................................................................ 37 4.1 Bonds: the bascs......................................................... 37 4.1.1 The prce/yeld relatonshp.......................................... 37 4.1.2 Floatng rate notes.................................................. 38 4.2 A general valuaton model................................................. 38 4.3 Interest rate contracts...................................................... 39 4.3.1 Lbor rates......................................................... 39 4.3.2 Swap rates......................................................... 4 4.3.3 Interest rate optons................................................. 4 4.3.4 Futures............................................................ 41 4.4 Prcng nterest-rate optons................................................ 42 4.4.1 The forward measure................................................ 42 4.4.2 Forwards and futures................................................ 44 4.4.3 Caplets............................................................ 44 4.4.4 Swaptons.......................................................... 45 References....................................................................... 47

1 Further Results n Stochastc Analyss 1.1 The Martngale Representaton Theorem for Brownan Moton Let W t, t be a Brownan moton on a probablty space (Ω, F, P ), and let F t be the natural fltraton: F t = σ{w s, s t}. Theorem 1.1. Let T > and suppose that X L 2 (Ω, F T, P ). Then there exsts an adapted process g t such that E T g2 (s)ds < and T X = EX + g(s)dw s. (1.1) The proof follows from the Lemmas below. Frst, recall that a subset D of L 2 (Ω, F T, P ) s dense f for every X L 2 (Ω, F T, P ) we have D B for every neghbourhood B of X. In partcular, there exsts a sequence X n D such that X n X. Lemma 1.2. Theorem 1.1 holds f the representaton (1.1) holds for every X n some dense subset D of L 2 (Ω, F T, P ). Proof: Let X L 2 (Ω, F T, P ) and take X n D, X n X as descrbed above. Then EX n EX and there exst ntegrands g n such that T X n = EX n + g n (s)dw s. (1.2) Takng X n = X n EX n we have the Ito sometry E( X n X m ) 2 = E T (g n (s) g m (s)) 2 ds (1.3) Snce X n s convergent, t s a Cauchy sequence, and hence from (1.3) the sequence g n s convergent n L 2 (Ω [, T ], dp dt). Thus there exsts g such that E T and (1.1) holds wth ths ntegrand g. (g n (s) g(s)) 2 ds as n Let D T be the subset of L 2 (Ω, F T, P ) consstng of random varables X of the form X = h(w t1, W t2,... W tn ), where n s an nteger, h s a bounded contnuous functon from R n to R, and t 1 <... < T n T. The proof of the followng result s an elegant applcaton of the martngale convergence theorem. See Øksendal [7], Lemma 4.3.1. Lemma 1.3. D T s dense n L 2 (Ω, F T, P ).

2 1 Further Results n Stochastc Analyss To prove the Theorem, t remans to show that any X D T has the representaton property, and ths we can show by a drect argument. In the followng, we take n = 2; the extenson to n > 2 s obvous. Frst, a fact about condtonal expectaton. Lemma 1.4. Let X, Y be random varables takng values n R n, R m respectvely, on a probablty space (Ω, F, P ). Let G be a sub-σ-feld of F, and suppose that X s ndependent of G whle Y s G-measurable. Then for any measurable functon f : R n+m R such that E f(x, Y ) <, we have E[f(X, Y ) G] = b(y ), where b(y) = f(x, y)μ X (dx). R n Here μ X s the dstrbuton of X, the measure on the Borel sets B n of R n defned by μ X (B) = P (X B) for B B n. Proof: We have to show that for all bounded real-valued G-measurable random varables Z we have E[Zf(X, Y )] = E[Zb(Y )]. Let μ X,Y,Z be the dstrbuton of the R n+m+1 -valued r.v. (X, Y, Z). Snce X s ndependent of G, the random varables X and (Y, Z) are ndependent, so that μ X,Y,Z (dx, dy, dz) = μ X (dx)μ Y,Z (dy, dz). Hence E[Zf(X, Y )] = zf(x, y)μ X,Y,Z (dx, dy, dz) ( ) = z f(x, y)μ X (dx) μ Y,Z (dy, dz) = zb(y)μ Y,Z (dy, dz) = E[Zb(Y )]. Lemma 1.5. Let h : R 2 R be a bounded contnuous functon and let t 1, t 2, t satsfy t 1 t t 2. Then E[h(W t1, W t2 ) F t ] = v 1 (t, W t1, W t ), where v 1 (t, x, y) = 1 /2(t h(x, z) 2 t) dz. (1.4) 2π(t2 t) e(z y)2 Proof: Wrtng h(w t1, W t2 ) = h(w t1, (W t2 W t ) + W t ), ths follows mmedately from Lemma 1.5, on takng X = W t2 W t1, Y = (W t1, W t ) R 2 and f(x, y) = h(y 1, x + y 2 ), and recallng that X N(, t 2 t). Lemma 1.6. The random varable X = h(w t1, W t2 ), as defned n Lemma 1.5, has the representaton property. Proof: It can be checked drectly from (1.4) that the functon v 1 satsfes v 1 t (t, x, y) + 1 2 and v 1 (T, x, y) = h(x, y). Hence by the Ito formula 2 v 1 (t, x, y) = y2 h(w t1, W t2 ) = v 1 (T, W t1, W t2 ) = v 1 (t 1, W t1, W t1 ) + t2 t 1 v 1 y (s, W t 1, W s )dw s, (1.5)

1.2 Changes of Measure 3 and we know from Lemma 1.5 that v 1 (t 1, W t1, W t1 ) = E[h(W t1, W t2 ) F t1 ]. Now defne v (t 1, x) = v 1 (t 1, x, x), and, for t < t 1 v (t, x) = 1 /2(t v (t 1, z) 1 t) dz. (1.6) 2π(t1 t) e(z x)2 As above, we have and the Ito formula gves v t (t, x) + 1 2 2 v 1 (t, x) =, x2 v (t 1, W t1 ) = v 1 (t 1, W t1, W t1 ) = v (, ) + t1 From (1.5),(1.7) we now see that t2 h(w t1, W t2 ) = v (, ) + g(s)dw s, where and that Ths completes the proof. g(s) = { ( v / y)(s, W s ), s < t 1 ( v 1 / y)(s, W t1, W s ), t 1 s < t 2, v (, ) = E[h(W t1, W t2 )]. v y (s, W s)dw s. (1.7) 1.2 Changes of Measure 1.2.1 Normal dstrbutons A random varable X s normally dstrbuted, wrtten X N(μ, σ 2 ), f ts characterstc functon ψ takes the form ψ μ (u) = Ee ux = exp (uμ 12 ) u2 σ 2. (1.8) Ths corresponds to the densty functon φ gven by ( 1 φ μ (x) = exp 1 ) (x μ)2. 2πσ 2 2σ2 μ and σ are the mean and standard devaton respectvely. (σ s fxed n the followng and so s not ncluded n the notaton.) If X N(μ, σ 2 ) then for any bounded functon f, E[f(X)] = f(x)φ μ (x)dx, For any ν we can trvally wrte ths as E[f(X)] = f(x) φ μ(x) φ ν (x) φ ν(x)dx, (1.9) and we fnd that ( ) φ μ (x) 1 φ ν (x) = exp 1 (μ ν)x σ2 2σ 2 (μ2 ν 2 ). (1.1) Let us denote by Λ the random varable Λ = φ μ (X)/φ ν (X). We fnd that

4 1 Further Results n Stochastc Analyss Λ >, E ν [Λ] = 1 E μ [f(x)] = E ν [f(x)λ], where E μ denotes ntegraton wrt N(μ, σ 2 ) To see the frst of these, take f(x) 1 n (1.9), or use (1.1) and the fact that f X N(ν, σ 2 ) then Ee X = e ν+ 1 2 σ2. We can thus flp between E μ and E ν by ntroducng Λ, the lkelhood rato or Radon-Nkodym dervatve. In most applcatons, ν =. 1.2.2 A General Settng Let (Ω, F, P ) be a probablty space, and Λ be a r.v. such that Λ a.s. and EΛ = 1. Then we can defne a measure Q on (Ω, F) by QF = ΛdP, F F. (1.11) F Λ s often wrtten dq/dp and s the Radon-Nkodym dervatve of Q wrt P. Note that P F = QF = ; we say that Q s absolutely contnuous wrt P, wrtten Q P. The Radon-Nkodym theorem states that any Q that s absolutely contnuous wrt P can be wrtten as (1.11) for some Λ. If Λ > a.s. then P s absolutely contnuous wrt Q, wth RN dervatve dp/dq = 1/Λ. In ths case P and Q are sad to be equvalent, wrtten P Q. Measures P and Q are equvalent f and only f they have the same null sets: P F = QF =. 1.2.3 Condtonal Expectatons Let X be an ntegrable r.v. and G a sub-sgma-feld of F. The condtonal expectaton of X gven G s the unque G-measurable r.v., denoted E[X G] such that XdP = E[X G]dP. Key propertes: G 1. E[X G] = X f X s G-measurable 2. E[X G] = EX f X s ndependent of G 3. E[Y X G] = Y E[X G] f Y s G-measurable 4. For H G, E[X H] = E[E[X G] H]. In partcular, EX = E(E[X G]) for any sub-σ-feld G. Exstence of E[X G] follows from the Radon-Nkodym theorem. Indeed, the formula Q(A) = A XdP defnes a measure on (Ω, G) that s absolutely contnuous wrt P, the restrcton of P to G. Hence there exsts a G-measurable functon Λ such that Q(A) = A ΛdP. The followng result wll be needed n Secton 1.3.3 below. Lemma 1.7. Suppose X, X 1, X 2,... s a sequence of ntegrable random varables such that X n X n L 1. Then for any σ-feld G, E[X n G] E[X G] n L 1. Proof: Frst we show that f Y s any ntegrable r.v. then G E[Y G] E[ Y G] a.s. (1.12) Indeed, denotng as usual Y + = max(y, ) and Y = Y + Y, we have E[Y G] + = E[Y + Y G] + E[Y + G] + = E[Y + G]

and E[Y G] = E[ Y G] + E[( Y ) + G] = E[Y G], from whch (1.12) follows. Now f X n X n L 1 then usng (1.12) E E[X n G] E[X G] = E E[X n X G] Condtonal expectaton under change of measure E (E[ X n X G]) = E X n X. 1.2 Changes of Measure 5 If P, Q are measures on (Ω, F) such that Q P wth RN dervatve Λ = dq/dp, and G s a sub-sgma-feld of F then E Q [X G] = E[XΛ G] a.s. Q (1.13) E[Λ G] To see ths, calculate E[XΛ G] by takng a set G G and usng the above propertes of condtonal expectaton. We get E[XΛ G]dP = XΛdP G G = XdQ G = E Q [X G]dQ G = E Q [X G]ΛdP G = E Q [X G]E[Λ G]dP G Thus G ZdP = for all G G, where Z = E[XΛ G] E Q[X G]E[Λ G] s a G-measurable random varable. Hence Z = a.s. Ths gves (1.13) on notng that, by defnton, the set {ω : E[Λ G] = } has Q-measure. Changes of measure and martngales Take a probablty space (Ω, F, P ) equpped wth a fltraton (F t, t [, T ]). Assume for convenence that F = F T, and suppose there s another measure Q, defned by dq/dp = Λ, where Λ s a non-negatve r.v. wth EΛ = 1. An adapted process (X t ) s a martngale (under measure P ) f t s ntegrable and for s t X s = E[X t F s ] a.s. The man result we need s ths: a process Y t s a Q-martngale f and only f the process X t = Y t Λ t s a P -martngale, where Λ t = E[Λ F t ]. Ths follows from (1.13). Indeed, for s < t we have E Q [Y t F s ] = E[Y tλ F s ] E[Λ F s ] = E[Y tλ t F s ] Λ s If Y t s a Q-martngale the left-hand sde s equal to Y s, so that Y t Λ t s a martngale, whle f Y t Λ t s a martngale then the rght-hand sde s equal to Y s, showng that Y t s a Q-martngale. A process X t s a local martngale f there exsts a sequence of stoppng tmes τ n such that τ n a.s. and for each n the process X n t = X t τn s a martngale. It s also true that a process Y t s a Q-local martngale f and only f the process X t = Y t Λ t s a P -local martngale. Exercse: show ths.

6 1 Further Results n Stochastc Analyss 1.3 The Lévy characterzaton of Brownan Moton 1.3.1 Quadratc varaton of Brownan moton Let W t be a Brownan moton process and let T be a fxed tme. For n = 1, 2,... let {t n, =..k n} be an ncreasng sequences of tmes wth t n =, tn k n = T. Denote ΔW = W t n +1 W t n,δt = t n +1 tn and S n = ΔW 2. Note that the r.v. ΔW are ndependent wth EΔW =, EΔW 2 = Δt. Hence that ES n = T and ESn 2 = 2 Δt 2 + T 2. (1.14) The latter follows from a short calculaton usng the fact that f X N(, σ 2 ) then EX 4 = 3σ 4. From (1.14), var(s n ) = E(S n T ) 2 = 2 Δt 2 2 max{δt } ΔT = 2T max{δt }. (1.15) Hence S n T n L 2 as n as long as max {Δt }. Let us now specalze to the case t n = /2n. From (1.15) and the Chebyshev nequalty, for any ɛ > 2T 2 n P [ S n T ) > ɛ] ɛ 2. Takng ɛ = 1/n we fnd that [ P S n T ) > 1 ] 2T n 2 2 n < n n n Hence by the Borel-Cantell lemma we have P [ S n T > 1n ] nfntely often =, showng that S n T almost surely. Thus for each T > the quadratc varaton QV(T ) s equal to the determnstc functon QV(T ) = T. Suppose now that X t s a contnuous process wth sample paths of bounded varaton,.e. sup X t n +1 X t n < a.s. n For example, any process of the form X t = t φ(s)ds wth ntegrable φ satsfes ths. Let us compute the quadratc varaton of Y t = W t + X t. We have (Y t n +1 Y t n ) 2 = (W t n +1 W t n + X t n +1 X t n ) 2 = ΔW 2 + ΔX 2 + 2 ΔW ΔX where ΔW = W t n +1 W t n to : for the thrd term, etc. The frst term converges to T and the second and thrd converge

(W t n +1 W t n )(X t n +1 X t n ) max 1.3 The Lévy characterzaton of Brownan Moton 7 W t n +1 W t n X t n +1 X t n. The sum on the rght s bounded and the max converges to zero because W t s a contnuous functon. A smlar argument apples to the second term. We have shown that the quadratc varaton of W and Y are the same: the quadratc varaton of W s not altered by addng a bounded varaton perturbaton to the sample path. 1.3.2 Quadratc varaton of contnuous martngales We can t treat ths subject n complete detal here; see [8] pages 52-55 or [2]. Let M t be a martngale on a fltered probablty space (Ω, F, (F t ), P ). Because of the martngale property, E[(M t M s ) 2 F s ] = E[M 2 t + M 2 s 2M t M s F s ] = E[M 2 t M 2 s F s ]. (1.16) and hence wth the notaton above [ ] [ ( ) ] E (M t n +1 M t n ) 2 = E E (M t n +1 M t n ) 2 Ft n = EMT 2, (1.17) usng (1.16). Ths suggests that the left-hand sde has a lmt as n, the quadratc varaton of (M t ). When (M t ) s Brownan moton we have from (1.16) for t > s E[M 2 t F s ] = E[M 2 t M 2 s F s ] + M 2 s = E[(M t M s ) 2 F s ] + M 2 s = t s + M 2 s. Hence the process M 2 t t s a martngale. The general stuaton s as follows. Theorem 1.8. Let M t be a contnuous local martngale. Then there s a unque contnuous ncreasng process, denoted [M] t, such that M 2 t [M] t s a local martngale. [M] t s the quadratc varaton of M t : t s the almost sure lmt of approxmatng sums as n (1.17) taken along sutable sequences (t n ). We call a process X t a semmartngale f t can be expressed as X t = M t + A t where M t s a martngale and A t s a process whose sample paths have bounded varaton. X t s a contnuous semmartngale f both M t and A t have contnuous sample paths, and a local semmartngale f M t s a local martngale. The exstence of [M] t gves us an Ito formula for contnuous local semmartngales, analogous to the usual Ito formula for Brownan moton. Theorem 1.9. Let X t be a contnuous local smmartngale and f a C 1,2 functon. Then 1.3.3 The Lévy characterzaton df(t, X t ) = f f dt + t x dx t + 1 2 f 2 x 2 d[m] t (1.18) Theorem 1.1. Let M t be a contnuous local martngale on a fltered probablty space (Ω, F, (F t ), P ), and suppose that [M] t = t, t. Then M t s an F t -Brownan moton. Proof: Suppose M t s a contnuous local martngale wth [M] t = t and take f(t, x) = exp(ux + u 2 t/2). By applyng (1.18) to the real and magnary parts of f you can check that (1.18) s also vald for complex functons. We obtan

8 1 Further Results n Stochastc Analyss df(t, M t ) = 1 2 u2 f(t, M t )dt + uf(t, M t )dm t 1 2 u2 f(t, M t )d[m] t, so that f(t, M t ) s a local martngale f [M] t = t. Thus for t > s we have ] E [e um t τn + 1 2 u2 t τ n Fs = e um s τn + 1 2 u2 s τ n, (1.19) where τ n s a sequence of localzng tmes. Now the sequence exp(um s τn + 1 2 u2 (s τ n )) s bounded and converges almost surely (and hence n L 1 ) to exp(um s + 1 2 u2 s). By Lemma 1.7, the condtonal expectaton n (1.19) converges n L 1 to the condtonal expectaton of the lmt, and we conclude that [ ] E e um t+ 1 2 u2 t F s = e um s+ 1 2 u2s, or, equvalently, [ E e u(m t M s ) ] F s = e 1 2 u2 (t s). (1.2) Now let Y be any F s -measurable random varable, and ψ Y be the characterstc functon of Y. Then by Property (3) of condtonal expectaton (see Secton 1.2.3 above) the jont characterstc functon of Y and M t M s s ψ Y,Mt M s (v, u) = E [e ] (vy +u(m t M s )) [e ] vy e u(m t M s ) = E = E [ [ e vy E e u(m t M s ) = E [ e vy ] e 1 2 u2 (t s) = ψ Y (v)ψ Mt M s (u). ]] F s Thus Y and (M t M s ) are ndependent, mplyng snce Y s arbtrary that (M t M s ) s ndependent of F s. From (1.2), (M t M s ) s normally dstrbuted wth mean and varance t s. Hence (M t ) s an (F t ) Brownan moton. The vector case In Chapter 3 we wll need a vector verson of the Lévy characterzaton. Thus, let M t = (Mt 1,..., Mt n ) be an n-vector process each of whose components Mt s a contnuous local martngale on a fltered probablty space (Ω, F, (F t ), P ). Note that for any two numbers a, b we have ab = 1 4 ((a + b)2 (a b) 2 ). (Ths s sometmes called the polarzaton formula.) Wth ths n mnd, defne [M, M j ] t = 1 ( [M + M j ] t [M M j ) ] t. 4 The processes [M + M j ], [M M j ] are the quadratc varaton processes of the local martngales M + M j and M M j respectvely, as ntroduced n Theorem 1.8 above. [M, M j ] s sometmes called the cross-varaton of the local martngales M, M j. Note that [M, M ] = [M ]. We can wrte the cross-varaton as a symmetrc n n matrx, denoted [M] t, wth, j th component [M, M j ]. We leave t as an exercse for the reader to show that ths matrx s almost surely nonnegatve defnte. The vector verson of the Ito formula (1.18) for contnuous local semmartngales s as follows. We start wth a vector of contnuous local semmartngales X t = M t +A t, = 1,..., n. Denote [X, X j ] = [M, M j ]. Then for any C 1,2 functon f : R + R n R, we have df(t, X t ) = f n t dt + f dx + 1 x 2 =1 n,j=1 2 f x x j d[x, X j ]. (1.21)

A partcularly useful specal case of (1.21) s the product formula: 1.4 The Grsanov Theorem 9 d(x 1 t X 2 t ) = X 1 t dx 2 t + X 1 t dx 2 t + d[x 1, X 2 ] t. (1.22) Theorem 1.11. Let M t be an R n -valued contnuous local martngale as descrbed above, and suppose that [M] t = It, where I denotes the n n dentty matrx. Then the components M t, = 1,..., n are ndependent Brownan motons. The proof of ths result s essentally the same as that of the unvarate case, Theorem 1.1 above. We show that under the condton stated, for any n-vector u and t > s, E [ e <u,m t M s > Fs ] = e 1 2 u 2 (t s). As before, ths shows that the ncrement M t M s s ndependent of F s wth dstrbuton N(, I(t s)). The concluson follows. 1.4 The Grsanov Theorem The Grsanov theorem states that, for Brownan moton, absolutely contnuous change of measure s equvalent to change of drft. Theorem 1.12. Let (Ω, F, {F t } t [,T ], P ) be a fltered probablty space, where < T < and we assume for convenence that F = F T. Let w t be an (F t, P )-Brownan moton. (a) Let g(t) be an adapted process satsfyng T g2 (s)ds < a.s. and defne ( T Λ T = exp g(s)dw s 1 ) T g 2 (s)ds. (1.23) 2 Suppose that E[Λ T ] = 1, and defne a measure Q on (Ω, F) by dq/dp = Λ T. Then under measure Q the process w t defned by w t = w t t g(s)ds s an F t Brownan moton. (b) Suppose F t s the natural fltraton of w t and that Q s a measure such that Q P. Then there exsts a process g(t) such that dq/dp s equal to Λ T defned by (1.23). Proof: (a) The assumpton that EΛ T = 1 ensures that Q s a probablty measure. Applyng the Ito formula, we fnd that d( wλ) = Λ( wg + 1)dw, so that wλ s a local martngale whch mples, as shown n secton 1.2.3, that w s a Q-local martngale. Certanly w has contnuous sample paths, and by the argument n secton 1.3.1 the quadratc varaton of w s equal to t. By the Lévy characterzaton, w s a Q-Brownan moton. (b) Let Q be an equvalent measure and defne Λ T = dq/dp. Then Λ T > a.s. and EΛ T = 1. For any t [, T ] let P t, Q t denote the restrctons of P and Q to F t. Then P t Q t and the Radon-Nkodym dervatve s dq t /dp t := Λ t = E[Λ T F t ]. Hence Λ t > a.s. By the martngale representaton theorem for Brownan moton, there exsts an ntegrand φ such that T φ2 (t)dt < and Λ t = 1 + Now apply the Ito formula to calculate t φ(s)dw s, t T. (1.24)

1 1 Further Results n Stochastc Analyss d log Λ t = 1 Λ t φ(t)dw t 1 2 1 φ 2 (t)dt. Thus Λ T s gven by (1.23) wth g(t) = φ(t)/λ t. Remarks (a) Let M t be a non-negatve local martngale,.e. for tmes τ n, for t > s M s τn = E[M t τn F s ]. Thus, usng Fatou s lemma for condtonal expectaton, Λ 2 t M s = lm nf n M s τn = lm nf n E[M t τn F s ] E[lm nf n M t τn F s ] = E[M t F s ]. Thus any non-negatve local martngale s a supermartngale, so that n partcular EM t s a decreasng functon of t. Now Λ T defned by (1.23) s a non-negatve local martngale, so the assumpton that EΛ T = 1 mples that EΛ t = 1 for all t [, T ], snce Λ = 1 a.s. (b) The best general suffcent condton mplyng EΛ T = 1 s the Novkov condton ( ) 1 T E exp g 2 (s)ds <. 2

2 The Black Scholes World 2.1 The Model To start wth we consder a world wth just one rsky asset wth prce process S t and a rsk-free savngs account payng constant nterest rate r wth contnuous compoundng. Everythng takes place n a fnte tme nterval [, T ]. Let (Ω, F, (F t ) t [,T ], (w t ) t [,T ] ) be Wener space,.e. w t s Brownan moton, F t s the natural fltraton of w t and F = F t. The prce process S t s supposed to be geometrc Brownan moton: S t satsfes the SDE ds t = μs t dt + σs t dw t (2.1) for gven drft μ and volatlty σ. (2.1) has a unque soluton: f S t satfes (2.1) then by the Ito formula d log S t = (μ 1 2 σ2 )dt + σdw t, so that S t satsfes (2.1) f and only f S t = S exp((μ 1 2 σ2 )t + σw t ). (2.2) Note that ths makes S t very easy to smulate: for any ncreasng sequence of tmes = t < t 1..., = S t 1 exp ((μ 12 σ2 )(t t 1 ) + σ ) t t 1 X, S t where X 1, X 2,... s a sequence of ndependent N[, 1] random varables. Ths representaton s exact. Another thng that follows from (2.2) s that ES t = S e μt. In the nterests of symmetry we want the savngs account also to be expressed as a traded asset,.e. we should nvest n t by buyng a certan number of unts of somethng. A convenent somethng s a zero-coupon bond B t = exp( r(t t)). Ths grows, as requred, at rate r: db t = rb t dt (2.3) Note that (2.3) does not depend on the fnal maturty T (the same growth rate s obtaned from any ZC bond) and the choce of T s a matter of convenence as we wll see below. 2.2 Portfolos and Tradng Strateges If we hold φ and ψ unts of S and B respectvely at tme t then we have a portfolo whose tmet value s φs t + ψb t. The assumptons of Black-Scholes are that we have a frctonless market,

12 2 The Black Scholes World meanng that S and B can be traded n arbtrary amounts wth no transacton costs, and short postons are allowed. In partcular ths means we can nvest n, or borrow from, the rskless account at the same rate r of nterest. A tradng strategy s then a trple (φ t, ψ t, x ), where x s the ntal endowment and (φ t, ψ t ) s a par of adapted processes satsfyng T The gan from trade n [s, t] s then φ 2 t dt < a.s., T ψ t dt <. t φ u ds u + t s s ψ u db u. (Note how ths matches up wth the defnton of the Ito ntegral!) A portfolo s self-fnancng f φ t S t + ψ t B t φ s S s ψ s B s = t φ u ds u + t s s The ncrease n portfolo value s entrely due to gans from trade. ψ u db u. 2.3 Arbtrage and Valuaton Denote by V t the portfolo value at tme t,.e. V t = φ t S t + ψ t B t. An arbtrage opportunty s the exstence of a self-fnancng tradng strategy and a tme t such that V =, V t a.s. and P [V t > ] > (or, equvalently, EV t >.) It s axomatc that arbtrage cannot exst n the market, so no mathematcal model should permt arbtrage opportuntes. 2.3.1 Forwards Consder a forward contract n whch we fx a prce K now to be pad at tme T for delvery of 1 unt of S T. The unque no-arbtrage value of K s F = e rt S. Indeed, suppose someone offers us a forward contract at K < F. We sell one share and nvest the proceeds S n the bank. At tme T we get the share back for a payment of K but the value of our bank account s F > K. We make a rskless proft of F K. If we are able to offer a forward at K > F then we should borrow S and buy the share. Agan, there s a rskless proft at tme T. (Ths argument s ndependent of the prcng model for S t.) 2.3.2 Put-Call Party A call opton wth strke K and exercse tme T has exercse value [S T K] +, and a put has exercse value [K S T ] +. Clearly [S T K] + [K S T ] + = S T K, so that buyng a call and sellng a put at tme zero s equvalent to buyng a forward and agreeng to pay K at tme T. Thus whatever the prces C and P at tme, they must satsfy C P = (F K)B. Note agan that ths s completely model-ndependent.

2.4 Black-Scholes: the Orgnal Proof 13 2.3.3 Replcaton Suppose there s a contngent clam wth exercse value h(s T ) at tme T (for example a put or call opton) and there exsts a self-fnancng tradng strategy (φ, ψ, x ) such that V T = h(s T ) a.s. Then x s the unque no-arbtrage prce of the contngent clam. The arbtrage, f avalable, s realzed by sellng the contngent clam and gong long the replcatng portfolo, or vce versa. Ths argument s sometmes known as the law of one prce: f two assets have dentcal cash flows n the future then they must have the same value now. 2.4 Black-Scholes: the Orgnal Proof Black s and Scholes orgnal proof of the famous formula [1] was a very drect argument showng that a replcatng portfolo exsts for the European call opton. Here s a verson of that argument. The dea s to assume a whole lot of thngs and then show they are all true. The frst assumpton s that there s a smooth functon C(t, S) such that the call opton has a value C(t, S t ) at tme t < T, wth lm t T C(t, S) = [S K] +. Suppose we form a portfolo n whch we are long one unt of the call opton and short a self-fnancng portfolo (φ, ψ, C(, S )). The value of ths portfolo at tme t s then X t = C(t, S t ) φ t S t ψ t B t, wth, n partcular, X =. By the Ito formula and the self-fnancng property, dx t = C ( C S ds + t + 1 2 σ2 St 2 2 ) C S 2 dt φ t ds t ψ t db t. If we choose φ t = C/ S and use the fact that db = rbdt we see that ( C dx t = t + 1 2 σ2 St 2 2 ) C S 2 dt ψ trb t dt. Let us now choose ψ t = C t + 1 2 σ2 St 2 2 C S 2, rb t herocally assumng that n dong so we have not destroyed that self-fnancng property. Then X t, so that C = φs + ψb = S C C S + t + 1 2 σ2 St 2 2 C S 2, r showng that C must satsfy the Black-Scholes PDE wth boundary condton C t C + rs S + +1 2 σ2 St 2 2 C rc = (2.4) S2 C(T, S) = [S K] +. (2.5) Equatons (2.4),(2.5) are enough to determne the functon C, as we wll show below. Is (φ, ψ, x ) n fact self-fnancng? By defnton φs + ψb = C (snce X t ) and t φds + t ψdb = = t t C S ds + dc t = C(t, S t ) C(, S ). ( C t + 1 2 σ2 St 2 2 ) C S 2 du

14 2 The Black Scholes World Ths confrms the self-fnancng property. We have now shown that the call opton can be replcated by a self-fnancng portfolo wth ntal endowment C(, S ), so by the argument n secton 2.3.3 ths s the unque arbtrage-free prce. In the above argument, no specal role s played by the call opton exercse functon [S T K] +. It smply provdes the boundary condton for the Black-Scholes PDE (2.4). If we used another boundary condton C(T, S) = h(s) then the correspondng soluton of (2.4) would gve us the no-arbtrage value and hedgng strategy for a contngent clam wth exercse value h(s T ). Example: the Forward Prce. It s easy to check that C(t, S) = S satsfes (2.4) wth boundary condton C(T, S) = S. For a constant K, C(t, S) = e r(t t) K satsfes (2.4) wth boundary condton C(T, S) = K. Snce (2.4) s a lnear equaton, the value at tme of recevng S T K at tme T s therefore S Ke rt, whch s equal to zero when K = e rt S, the forward prce. You can check (please do!) that the hedgng strategy φ = C/ S mpled by Black-Scholes concdes wth the strategy gven n secton 2.3.1. 2.4.1 Probablstc soluton of the Black-Scholes PDE Suppose we have an SDE dx t = m(x t )dt + g(x t )dw t where m, g are Lpschtz contnuous functons so that a soluton exsts. The dfferental generator of x t s the operator A defned by so the Ito formula can be wrtten Af(x) = m(x) f x + 1 2 g2 (x) 2 f x 2 df(x t ) = Af(x t )dt + f x g(x t)dw t. Now consder the followng PDE for a functon v(t, x) v + Av(t, x) rv(t, x) =, t < T, (2.6) t v(t, x) = Ξ(x), (2.7) where r s a gven constant and Ξ a gven functon. If v satsfes ths then applyng the Ito formula we fnd that ( v d(e rt v(t, x t )) = re rt v(t, x t )dt + e rt t + Av(t, x t)dt + v ) x g(t, x t)dw t rt v = e x g(t, x t)dw t. (2.8) Thus exp( rt)v(t, x t ) s a local martngale. If t s a martngale then ntegratng from t to T and usng (2.7) we see that ] v(t, x t ) = E t,x [e r(t t) Ξ(x T ) Ths s the probablstc representaton of the soluton of the PDE (2.6),(2.7). Comparng (2.4),(2.5) wth (2.6),(2.7) we see that these equatons match up when m(x) = rx and g(x) = σx,.e. x t satsfes dx t = rx t dt + σx t dw t. (2.9)

Now return to the prce model (2.1) and ntroduce a measure change ( dq dp = exp αw T 1 ) 2 α2 T, 2.5 Proof by Martngale Representaton 15 where α s a constant. By Grsanov, d ˇw = dw αdt s a Q-Brownan moton, n terms of whch (2.1) becomes ds t = μs t dt + σs t (d ˇw t + α dt). Choosng α = (r μ)/σ we get ds t = rs t dt + σs t d ˇw t, (2.1) the same equaton as (2.9). Equaton (2.1) s the prce process expressed n the rsk-neutral measure Q, and the above argument shows that the probablstc soluton of the Black-Scholes PDE (2.4),(2.5) s ( C(t, S) = E Q t,s e r(t t) [S T K] +). Ths s however easly computed snce S T s gven explctly n terms of w T by (2.2) (wth r replacng μ). We get C(t, S) = e r(t t) 2π A short calculaton gves the fnal expresson [S exp((r σ 2 /2)(T t) σx T t) K] + e x2 /2 dx. C(t, S) = SN(d 1 ) e r(t t) KN(d 2 ) (2.11) where N( ) denotes the cumulatve standard normal dstrbuton functon and d 1 = log(s/k) + (r + σ2 /2)(T t) σ T t d 2 = d 1 σ T t We can now te up the loose ends of the argument. It can be checked drectly that C defned by (2.11) does satsfy the Black-Scholes PDE (2.4),(2.5), and another short calculaton (see Problems II!) shows that C/ S = N(d 1 ), so that n partcular < C/ S < 1. Hence the ntegrand n (2.8) s square-ntegrable and the stochastc ntegral s a martngale, as requred. Another verson of the formula, often more useful, s ths. Recall that the forward prce at t for delvery at T s F = Se r(t t). We can therefore express (2.11) as C(t, S) = e r(t t) (F N(d 1 ) KN(d 2 )), (2.12) and d 1 can be expressed as d 1 = log(f/k) + 1 2 σ2 (T t) σ. T t 2.5 Proof by Martngale Representaton Let φ be an adapted process wth and let X t be a process defned by T φ 2 t S 2 t dt < a.s. (2.13) dx t = φ t ds t + (X t φ t S t )r dt, X = x. (2.14)

16 2 The Black Scholes World The nterpretaton s that X t s the portfolo value correspondng to the tradng strategy (φ, ψ, x ) where ψ t = X t φ t S t, (2.15) B(t).e. φ t unts of the rsky asset are held and the remanng value X t φ t S t s held n the savngs account. Ths strategy s always self-fnancng snce X t s by defnton the gans from trade process, whle the value s φs + ψb = X. Applyng the Ito formula we fnd that, wth X t = e rt X t, d X t = φe rt S((μ r) dt + σ dw) (2.16) = φ Sσd ˇw, (2.17) where S t = e rt S t. (The frst lne (2.16) shows ncdentally that (2.14) has a unque soluton.) Thus e rt X t s a local martngale n the rsk-neutral measure Q. Suppose we have an opton whose exercse value at tme T s H, where H s an F T -measurable random varable wth EH 2 <. By the martngale representaton theorem there s an ntegrand g such that T e rt H = E Q [e rt H] + g t d ˇw t. Defne φ t = e rt g t /(σs t ) and then ψ by (2.15) and x = E Q [e rt H]. Then φ Sσ = g and the tradng strategy (φ, ψ, x ) generates a portfolo value process such that X T = H a.s.,.e. (φ, ψ, x ) s a replcatng portfolo for H. It follows that the opton value s x = E Q [e rt H]. Note ths s a much more general result than that obtaned by the prevous argument, n that the opton payoff can be an arbtrary, possbly path-dependent, random varable, whereas before we assumed t took a value of the form H = h(s T ). On the other hand the above argument only asserts that a replcatng portfolo exsts: t does not gve an explct formula for φ. Theorem 2.1. Let Φ be the class of nvestment strateges φ t such that (a) the ntegrablty condton (2.13) s satsfed, and (b) there exsts a postve constant A φ such that X t A φ for all t [, T ], where X t s the process defned by (2.14). In the Black-Scholes model, no strategy φ Φ s an arbtrage opportunty. Proof: Suppose X t s gven by (2.14) for some strategy φ Φ and X =. Then, from (2.17), the dscounted process X t s a Q-local martngale whch s bounded below by the constant A φ. Thus X t + A φ s a non-negatve local martngale, and hence a supermartngale. Therefore X t s a supermartngale and has decreasng expectaton: for any t > = E Q [ X ] E Q [ X t ]. (2.18) On the other hand, f X t a.s.(p ) and P [X t > ] > then, snce P and Q are equvalent measures, E Q [ X t ] >, whch s ncompatble wth (2.18). Hence there cannot be an arbtrage opportunty as defned n Secton 2.3. 2.6 Robustness of Black-Scholes Hedgng If we assume the Black-Scholes prce model (2.1) then the prce at tme t of an opton wth exercse value h(s T ) s C h (S t, r, σ, t) = C(t, S t ) where C(t, S) satsfes the Black-Scholes PDE (2.4) wth boundary condton C(T, S) = h(s).

2.6 Robustness of Black-Scholes Hedgng 17 Suppose we sell an opton at mpled volatlty ˆσ,.e. we receve at tme the premum C h (S, r, ˆσ, ), and we hedge under the assumpton that the model (2.1) s correct wth σ = ˆσ. The hedgng strategy s then delta hedgng : the number of unts of the rsky asset held at tme t s the so-called opton delta C/ S: φ t = C S (t, S t). (2.19) Suppose now that the model (2.1) s not correct, but the true prce model s ds t = α(t, ω)s t dt + β(t, ω)s t dw t, (2.2) where w t s an F t -Brownan moton for some fltraton F t (not necessarly the natural fltraton of w t ) and α t, β t are F t -adapted, say bounded, processes. It s no loss of generalty to wrte the drft and dffuson n (2.2) as αs, βs: snce S t > a.s. we could always wrte a general dffuson coeffcent γ as γ t = (γ t /S t )S t α t S t. In fact the model (2.2) s sayng lttle more than that S t s a postve process wth contnuous sample paths. Usng strategy (2.19) the value X t of the hedgng portfolo s gven by X = C(, S ) and dx t = C S ds t + ( X t C S S t ) r dt where S t satsfes (2.2). By the Ito formula, Y t C(t, S t ) satsfes dy t = C ( C S ds + t + 1 2 β2 St 2 2 ) C S 2 dt. Thus the hedgng error Z t X t Y t satsfes d dt Z C t = rx t rs t S C t 1 2 β2 St 2 2 C S 2. Usng (2.4) and denotng Γ t = Γ (t, S t ) = 2 C(t, S t )/ s 2, we fnd that Snce Z =, the fnal hedgng error s Comments: d dt Z t = rz t + 1 2 S2 t Γ 2 t (ˆσ 2 β 2 t ). Z T = X T h(s T ) = T e r(t s) 1 2 S2 t Γ 2 t (ˆσ 2 β 2 t )dt. Ths s a key formula, as t shows that successful hedgng s qute possble even under sgnfcant model error. It s hard to magne that the dervatves ndustry could exst at all wthout some result of ths knd. Notce that: Successful hedgng depends entrely on the relatonshp between the Black-Scholes mpled volatlty ˆσ and the true local volatlty β t. For example, f we are lucky and ˆσ 2 βt 2 a.s. for all t then the hedgng strategy (2.19) makes a proft wth probablty one even though the true prce model s substantally dfferent from the assumed model (2.1), as long as Γ t, whch holds for standard puts and calls. The hedgng error also depends on the opton convexty Γ. If Γ s small then hedgng error s small even f the volatlty has been underestmated.

18 2 The Black Scholes World 2.7 Optons on Dvdend-payng Assets Holders of ordnary shares receve dvdends, whch are cash payments normally quoted as x pence per share, pad on specfc dates wth the value x beng announced some tme n advance. For a stock ndex, where the consttuent stocks are all payng dfferent dvdends at dfferent tmes, t makes sense to thnk n terms of a dvdend yeld, the dvdend per unt tme expressed as a fracton of the ndex value. In mathematcal terms, we assume that a dvdend s a contnuous-tme payment stream, the dvdend pad n a tme nterval dt beng qs t dt. Thus q s the dvdend yeld. In ths secton we analyse the case where q s a fxed constant. Equaton (2.14), descrbng the evoluton of a self-fnancng portfolo, must be modfed to so that dx t = φ t ds t + qφ t S t dt + (X t φ t S t )r dt, X = x. (2.21) Now change to a martngale measure Q q such that = φ t S t (μ + q r)dt + X t r dt + φ t S t σdw t, (2.22) d ( e rt X t ) = φ S(μ + q r)dt + φ Sσ dw. (2.23) dw q = dw + μ + q r dt σ s a Q q -Brownan moton. Then (2.23) becomes smply d ( e rt X t ) = φ Sσ dw q. Thus by the argument of the prevous secton, the prce at tme of a contngent clam H s [ p = E Q q e rt H ]. (2.24) In partcular, take H = S T. Then p s the no-arbtrage prce now for delvery of 1 unt of the asset at tme T, or, equvalently, e rt p s the forward prce. In (2.21), take X φs =, so that all recepts are re-nvested n the rsky asset S, nothng beng held n the bank account. Then φ = X/S, so that dx = X S ds + q X S S dt = X(μ dt + σ dw) + qx dt = X((μ + q)dt + σ dw). (2.25) On the other hand, f we defne Ŝt = e qt S t and use (2.1) and the Ito formula, we fnd that dŝt = Ŝt((μ + q)dt + σ dw). (2.26) From (2.25) and (2.26) we see that X t = Ŝt = e qt S t for all t > f X = S. Now the soluton of (2.25) s lnear n the ntal condton, so f X = e qt S then X T = S T a.s. We have shown the followng. Proposton 2.2. () For an asset wth a constant dvdend yeld q, the forward prce at tme T s F T = e (r q)t S. The replcatng strategy that delvers one unt of the asset at tme T conssts of buyng e qt unts of the asset at tme and renvestng all dvdends n the asset. () The value of a call opton on the asset wth exercse tme T and strke K s C(S, K, r, q, σ, T ) = e rt (F T N(d 1 ) KN(d 2 )), (2.27) where d 1 = log(f T /K) + σ 2 T/2 σ, d 2 = d 1 σ T. T

2.8 Barrer Optons 19 Proof: Only part () remans to be proved. Under measure Q q, S t satsfes ds t = (r q)s t dt + σs t dw q t, (2.28) and the opton value s gven by (2.24) wth H = [S T K] +. Ths s exactly the same calculaton as standard Black-Scholes, but wth (r q) replacng r n the prce equaton (2.28) (but not n the dscount factor e rt n (2.24)). Formula (2.27) follows from (2.12). 2.8 Barrer Optons Let S t be a prce process and let M t = max u t S u be the maxmum prce to date. An up-and-out call opton has exercse value [S T K] + 1 MT <B. It pays the standard call payoff f S t < B for all t [, T ] and zero otherwse. B s the barrer, and to make sense, we must have S < B, K < B. An up-and-n call opton pays [S T K] + 1 MT B. The sum of these two payoffs s an ordnary call, so we only need to value one of the above. There are analogous defntons for down-and-out and down-and-n optons. Remarkably, there are analytc formulas for the values of these optons n the Black-Scholes world. These formulas but not the proofs can be found on pages 462-464 of Hull s book [5] The startng pont s the so-called reflecton prncple for Brownan moton. Let x t be standard Brownan moton startng at zero and m t = max s t x s. The reflecton prncple states that for y > and x y, ( ) x 2y P [m t y, x t < x] = N. (2.29) t The dea s that those paths that do ht level y before tme t restart from level y wth symmetrc dstrbuton (see fgure 2.1), so there s equal probablty that they wll be below x = y (y x) or above y + (y x) = 2y x at tme t. But P [m t y, x t 2y x] = P [x t 2y x] ( ) 2y x = 1 N t ( ) x 2y = N t Now [x t < x] = [m t < y, x t < x] [m t y, x t < x], so ( ) x N = P [x t < x] = P [m t < y, x t < x] + P [m t y, x t < x]. (2.3) t We have shown the followng. Proposton 2.3. The jont dstrbuton of x t, the Brownan moton at tme t, and ts maxmumto-date m t s gven by ( ) ( ) x x 2y F (y, x) = P [m t < y, x t < x] = N N (2.31) t t

2 2 The Black Scholes World 3 25 2 15 1 5-5 1 21 41 61 81 11 121 141 161 181-1 -15 Fg. 2.1. Reflecton prncple Ths argument depends on symmetry and doesn t work f x t has drft. We can use the Grsanov theorem to get the answer n ths case. If P ν denotes the probablty measure of BM wth drft ν (.e. x t = νt + w t where w t s ordnary BM) then we know that on the nterval [, T ] dp ν dp = exp(νx T 1 2 ν2 T ). Thus f f s any ntegrable functon then, usng (2.31), [ E ν [1 mt <yf(x T )] = E 1 mt <yf(x T ) dp ] ν dp ( = E [1 mt <yf(x T ) exp νx T 1 )] 2 ν2 T = y f(x)e (νx ν2 T/2) 1 (φ(x/ T ) φ((x 2y)/ ) T ) dx, T where φ(x) = e x2 /2 / 2π s the standard normal densty functon. Now clearly ( 1 exp 1 2πT 2T x2 + νx 1 ) 2 ν2 T = 1 ( ) x νt φ T T whle after some calculaton we fnd that 1 2πT exp ( 1 2T (x 2y)2 + νx 1 2 ν2 T ) ( ) = e2yν x 2y νt φ. T T Ths gves us the fnal result: the jont dstrbuton functon wth drft ν s ( ( ) ( )) x νt x 2y νt F ν (y, x) = P ν [m T < y, x T < x] = N e 2yν N. (2.32) T T Ths does concde wth F when ν =. A good reference for the above argument s Harrson [3]. Let us now return to barrer opton prcng. The prce process n the rsk-neutral measure s S T = S exp((r σ 2 /2)T + σw T ) whch we can wrte as where x T = w T + νt wth S T = S e σx T

ν = 1 σ (r 12 σ2 ). 2.8 Barrer Optons 21 The prce S T s n the money but below the barrer level when x T (a 1, a 2 ) where a 1 = 1 ( ) K σ log, a 2 = 1 ( ) B S σ log. S Denotng g(y, x) = F ν (y, x)/ x, the opton value can now be expressed as [ E ν e rt [S T K] + ] 1 MT <B = e rt a2 a 1 (S e σx K)g(y, x)dx. Dong the calculatons we obtan the opton value gven n [5] as a sum of four terms of the form c 1 N(c 2 ), as n the Black-Scholes formula. The up-and-out opton prce s ( ) 2λ B S (N(d 1 ) N(x 1 ) + (N( y) N( y 1 ))) S +Ke ( N(d rt 2 ) + N(x 1 σ ( ) 2λ 2 B T ) (N( y + σ T ) N( y 1 + σ T )) S where d 1, d 2 are the usual coeffcents and x 1 = log(s /B) σ T y 1 = log(b/s ) σ T + λσ T + λσ T y = log(b2 /(S K)) σ T λ = r + σ2 /2 σ 2 + λσ T Fgures 2.2,2.3,2.4 show the value, delta and gamma of an up-and-out call opton wth strke K = 1, barrer level B = 12 and volatlty 25%. The opton matures at tme T = 1. One can clearly see the black hole of barrer optons: the regon where the tme-to-go s short and the prced s close to the barrer. In ths regon there s hgh negatve delta, and there comes a pont where hedgng s essentally mpossble because of the large gamma (.e. unrealstcally frequent rehedgng s called for by the theory.)

18 22 2 The Black Scholes World 16 14 12 1 8 6 6 8 9 4 1 2 1 7 1 1 1 1 1 5 1 1 9. 5 Delta. 6. 7. 8. 9. 9 9 9 4 2-2 1. Fg. 2.2. Barrer opton value. -1. -2. -3. -4. -5.. 8 9. 9 2. 9 5. 9 8 1 1 8 1 1 5 1 1 2 1 9 1 6 1 2 9 6 9 7 5 6 Fg. 2.3. Barrer opton delta

.2. 2.8 Barrer Optons 23 -.2 -.4 -.6 -.8-1. -1.2. 8 9. 9 1. 9 3. 9 5. 9 7 1. 1 9 8 9 1 1 5 1 1 2 1 9 1 6 1 2 9 6 9 7 5 6 Fg. 2.4. Barrer opton gamma

3 Mult-Asset Optons Ths chapter covers prcng of optons where the exercse value depends on more than one rsky asset. Secton 3.1 descrbes a very useful formula for prcng exchange optons, whle Secton 3.2 gves a model for the FX market, where the opton could be drectly an FX opton or an opton on an asset denomnated n a foregn currency. Fnally, n Secton 3.3 we ntroduce the deas of numérare assets and changes of numérare. These deas gve extra nsght nto the exchange opton (and smpler calculatons) and play a bg role n nterest rate theory. 3.1 The Margrabe Formula Ths s an expresson, orgnally derved by Margrabe [6], for the value C(t, s 1, s 2 ) = E[e r(t t ) max(s 1 (T ) S 2 (T ), )] (3.1) of the opton to exchange asset 2 for asset 1 at tme T. It s assumed that under the rsk-neutral measure P, S 1 (t) and S 2 (t) satsfy ds 1 (t) = rs 1 (t)dt + σ 1 S 1 (t)dw 1, S 1 (t ) = s 1 (3.2) ds 2 (t) = rs 2 (t)dt + σ 2 S 2 (t)dw 2, S 2 (t ) = s 2, (3.3) where w 1, w 2 are Brownan motons wth E[dw 1 dw 2 ] = ρdt. The rskless rate s r. The Margrabe formula s C(t, s 1, s 2 ) = s 1 N(d 1 ) s 2 N(d 2 ) (3.4) where N( ) s the normal dstrbuton functon, The followng mportant facts can be noted rght away. 1. The solutons of (3.2), (3.3) are d 1 = ln(s 1/s 2 ) + 1 2 σ2 (T t ) σ T t (3.5) d 2 = d 1 σ T t (3.6) σ = σ1 2 + σ2 2 2ρσ 1σ 2 (3.7) S (t) = s e r(t t) M (t, t), = 1, 2, (3.8) where ( M (t, t) = exp σ (w (t) w (t )) 1 ) 2 σ2 (t t ), = 1, 2. (3.9) The process t M (t, t) s a martngale, and M (t, t ) = 1, so n partcular EM (t, t) = 1.

26 3 Mult-Asset Optons 2. The functon C defned by (3.1) does not depend on the rskless rate r. Indeed, from (3.8) we see that e r(t t ) s a factor of S 1 (T ) and S 2 (T ), and ths cancels the dscount factor e r(t t ) n (3.1). 3. The functon C s homogeneous of degree 1,.e. for any λ > we have C(t, λs 1, λs 2 ) = λc(t, s 1, s 2 ). (3.1) Ths s evdent from the defnton of the exercse value n (3.1) and the fact seen from (3.8) that S (T ) s lnear n the ntal condton s. Suppose that the functon C s contnuously dfferentable n s 1 and s 2. Then dfferentatng both sdes of (3.1) wth respect to λ and settng λ = 1 we obtan the key relatonshp 3.1.1 The Probablstc Method s 1 C s 1 + s 2 C s 2 = C (3.11) Snce C does not depend on the rskless rate r, we may and shall assume that r =. Wthout loss of generalty, we also take t =. Then C = E[max(S 1 (T ) S 2 (T ), )] [ ( )] S1 (T ) = E S 2 (T ) max S 2 (T ) 1, (3.12) By the Ito formula, Y (t) = S 1 (t)/s 2 (t) satsfes dy = Y (σ 2 2 σ 1 σ 2 ρ)dt + Y (σ 1 dw 1 σ 2 dw 2 ). (3.13) Now S 2 (t) = s 2 M 2 (, t) and we can regard M 2 (, T ) as a Grsanov exponental defnng a measure change d P dp = M 2(T ). (3.14) Thus from (3.12) C = s 2 Ẽ[max(Y (T ) 1, )] (3.15) where Ẽ denotes expectaton under measure P. By the Grsanov theorem, under measure P the process d w 2 = dw 2 σ 2 dt s a Brownan moton. We can wrte w 1 as w 1 (t) = ρw 2 (t)+ 1 ρ 2 w (t) where w (t) s a Brownan moton ndependent of w 2 (t) (under measure P ). It s shown below n Lemma 3.1 that w remans a Brownan moton under P, ndependent of w 2. Hence d w 1 defned by d w 1 = ρd w 2 (t) + 1 ρ 2 dw (t) = dw 1 (t) ρσ 2 dt s a P -Brownan moton. Usng (3.13), we fnd that the equaton for Y under P s whch we can wrte dy = Y (σ 1 d w 1 σ 2 d w 2 ) dy = Y σdw, (3.16) where w s a standard Brownan moton and σ s gven by (3.7). We wll see later n Secton 3.3 just why t s that Y s a P -martngale. In vew of (3.15), (3.16) the exchange opton s equvalent

3.1 The Margrabe Formula 27 to a call opton on asset Y wth volatlty σ, strke 1 and rskless rate. By the Black-Scholes formula, ths s (3.4). Remark: We can recover Black-Scholes from Margrabe smply by takng σ 2 = and s 2 = e r(t t) K; then S 2 (T ) = K a.s. Lemma 3.1. Suppose B t, B t are ndependent Brownan motons on a fltered probablty space (Ω, (F t ) t T, P ), and that φ t s an adapted process such that T φ2 sds < and EΛ(T ) = 1, where ( t Λ(t) = exp φ s db s 1 t ) φ 2 2 sds. Defne a measure P on (Ω, F T ) by takng d P /dp = Λ(T ), and a process B by d B t = db t φ t dt. Then B t, B t are ndependent Brownan motons under measure P. Proof. Ths uses the Grsanov theorem, Theorem 1.12, together wth the vector verson, Theorem 1.11, of the Lévy characterzaton theorem. From the Grsanov theorem we know that B t s a P -Brownan moton. Snce B, B are ndependent under P we fnd by applyng the Ito product formula (1.22) that ΛB s a P -local martngale and hence B s a P -local martngale. Its quadratc varaton s t so by the Lévy characterzaton B s a P -Brownan moton. To complete the proof we have to show that B, B are ndependent under P. Under P, the process B + B has quadratc varaton 2t. Snce B + B and B + B dffer by a process of bounded varaton, ths shows that [ B + B ] = 2t. By the same argument, [ B B ] = 2t, and hence [ B, B ] =. We have thus shown that the vector process M t = ( B t, B t) s a P local martngale wth cross-varaton process [M] t = It. Hence B t, B t are ndependent Brownan motons under P, by Theorem 1.11. 3.1.2 Hedgng an Exchange Opton Havng determned the value of the exchange opton, we now want to fnd the hedgng strategy that replcates ts exercse value. Note that there are n prncple three traded assets: S 1, S 2 and the zero-coupon bond P (t, T ). In fact, the Margrabe hedgng strategy only nvests n two of them: S 1 and S 2. We get some hnt of ths from the fact that the opton value does not depend on r; t s hard to magne how ths could be the case f hedgng were to nvolve the rskless asset. The key to ths queston s the homogenety property and specfcally the property (3.1) of the Margrabe value C. Recall that a tradng strategy s a trple of processes (α 1 (t), α 2 (t), α 3 (t)) whose values at tme t are the number of unts of S 1, S 2 and P, respectvely, held n the hedgng portfolo at tme t. Wth C equal to the Margrabe value (3.4), defne α 1 (t) = C s 1 (t, S 1 (t), S 2 (t)), α 2 (t) = C s 2 (t, S 1 (t), S 2 (t)), α 3 (t) =. (3.17) Then (3.1) states that C(t, S 1 (t), S 2 (t)) = α 1 (t)s 1 (t) + α 2 (t)s 2 (t), showng that ths strategy s automatcally replcatng snce n partcular C(T, S 1 (T ), S 2 (T )) concdes wth the Margrabe exercse value. We only need to show that ths strategy s self-fnancng whch, n vew of (3.17), s equvalent to showng that dc = α 1 ds 1 + α 2 ds 2. However, applyng the Ito formula, we have dc = α 1 ds 1 + α 2 ds 2 + C t + 1 2 σ2 1S1 2 2 C s 2 + 1 1 2 σ2 2S2 2 2 C 2 C s 2 + ρσ 1 σ 2 S 1 S 2 2 s 1 s 2 so a suffcent condton for the self-fnancng property s that C satsfes the PDE